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Constrained yielding in niobium single crystals bonded to sapphire
PII:
Acta mater. Vol. 46, No. 10, pp. 3571±3581, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00028-7 1359-6454/98 $19.00 + 0.00
CONSTRAINED YIELDING IN NIOBIUM SINGLE CRYSTALS BONDED TO SAPPHIRE G. SOYEZ1, G. ELSSNER1, M. RUÈHLE1 and R. RAJ2 Max-Planck-Institut fuÈr Metallforschung, D-70174 Stuttgart, Germany and 2University of Colorado, Boulder, CO 80309-0427, U.S.A.
1
(Received 20 October 1997; accepted 30 December 1997) AbstractÐThe mechanical properties of single crystalline sapphire/Nb/sapphire joints manufactured by diusion bonding in UHV are investigated for dierent orientation relationships (OR) between metal and ceramic at the interface. Compression tests resulted in the Young's moduli and the yield stresses of the joints for four dierent OR and metal thicknesses between 1 mm and 3 mm, respectively. The mechanical properties depend strongly on the thickness of the metal sheet, whereas the OR and hence the adhesion at the interface seem to have no signi®cant eect on the mechanical properties of the Nb/a-Al2O3 interface. The observed relationship between yield stress and metal thickness can be explained by a simple continuum mechanical model. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
The properties of metal/ceramic composites and joints are strongly in¯uenced by their heterophase interfaces. Interface failure depends on the elastic properties of the bonded materials, the thickness of the metal layer, the geometry of the specimens and the loading mode. Additionally, stresses caused by elastic and/or thermal mis®t lower the stability against mechanical loading. For optimized interfaces these residual stresses should be minimized. Information on the fracture energy and other measures of the bond strength prevail in the hitherto published studies on mechanical properties of metal/ceramic composites. These properties were investigated in general as a function of the material combination [1] and in the case of bicrystals as a function of the crystallographic orientation and the impurity content of the interface [2]. The dislocation mechanisms of plastic deformation near metal/ceramic interfaces however were not yet investigated. The experimental investigation of the mechanical behavior of metal/ceramic interfaces requires a wellde®ned preparation of the joints. In order to investigate the in¯uence of dierent OR at the interface on the mechanical properties, diusion bonding in an ultra high vacuum (UHV) was chosen for manufacturing. The model system niobium/sapphire was used since both materials possess a similar thermal expansion coecient and there are no intermediate reaction layers. Furthermore, the interface between Nb and sapphire is well characterized regarding structure [3±7] and mechanical properties [8±12]. In the present work the in¯uence of metal layer thickness, crystallographic orientation of the interface and fracture energy on the yield stress of the
metal part of the sapphire/Nb/sapphire sandwich was studied in compression tests. A second paper to be published will deal with the fracture mechanism of these sandwiches tested in compression. 2. SPECIMEN PREPARATION
High purity Nb and sapphire single crystals (impurity contents are listed in Table 1) were diusion bonded in UHV using a special apparatus described elsewhere [13]. To achieve optimum bonding conditions, the crystals have to be prepared in the following manner. After crystallographic orientation by Laue techniques, the surface planes of the crystals to be joined are prepared by grinding and mechanical polishing. The mean roughness of the polished surfaces is 10±20 nm for Nb and 8±15 nm for sapphire. Prior to inserting into the UHV bonding machine, the Nb is etched in concentrated HF. Afterwards both the metal and ceramic pieces are cleaned in acetone (p.a.) and methanol (p.a.) followed by ultrasonic agitation to remove remaining impurities from the polishing and etching process. The specimens are then dried in pure nitrogen (99.99 wt%). Before welding the surfaces to be bonded are sputter-cleaned by high-energetic Ar ions. The surface contamination was monitored by Auger electron spectroscopy. The stacking sequence of the sandwich con®guration and the subsequent preparation steps are shown in Fig. 1. Preliminary experiments showed, that stress concentrations between sapphire and alumina at the edges and at the contact areas, respectively, resulted in cracking of the sapphire during the bonding procedure. To avoid cracking,
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SOYEZ et al.: CONSTRAINED YIELDING Table 1. Impurity contents of the single crystals used
Nb Sapphire (a-Al2O3)
50 wt±ppm O2 R100 ppm Mg
27 wt±ppm N2 R20 ppm Ca
the sapphire/Nb/sapphire specimen was enclosed between two Mo platelets. Thus stress concentrations acting on the sapphire are reduced by the plasticity of the Mo. Furthermore, Mo does not bond with the Al2O3. The whole stack [Fig. 1(a)] is heated to a bonding temperature of 14008C within 1.5 h, held at this temperature for 3 h and then cooled down to room temperature in another 1.5 h. After reaching the bonding temperature of 14008C a pressure of about 7 MPa was applied to the stack of specimens. The single crystals in the stack were orientated so that the same OR holds at both Nb/sapphire interfaces, see Table 2. Dierent interface strengths, characterized by the fracture energy Gc in fourpoint bending, were realized by changing the orientation of the sapphire with respect to the Nb. In the ®rst three tri-crystal-combinations TCC1 to TCC3, the {110}Nb-plane is bonded onto sapphire planes with dierent OR. The identical orientation of the Nb should guarantee that the activated slip systems and hence the deformation mode should be the same for TCC1 to TCC3. TCC2 is consistent with the OR found in internally oxidized Nb±Alalloys [14]. TCC4 is dierent from the ®rst three
combinations since a (111)Nb-plane is bonded onto (0001)-sapphire substrates. TCC4 corresponds to the OR obtained for MBE grown overlayers on (0001) sapphire [15, 16]. The bonded stack [Fig. 1(b)] is withdrawn from the UHV diusion bonding machine and cut into several compression test specimens of the type shown in Fig. 1(c). The limited size of the samples that can be handled within the UHV diusion bonding machine and the cross section of the Nb single crystals determined the dimensions of the compression specimens. Up to four compression specimens could be cut out from one diusion bonded sample. They had a length between 5 mm and 8 mm and a quadratic cross section with a specimen width a 1 b of 3.8 2 0.3 mm. To modify the in¯uence of the interface strength on the mechanical properties, the thickness h of the Nb layer was varied between 1 mm and 3 mm. A parallel alignment of the top and bottom surfaces was attained by grinding. To measure the strain, two strain gauges were glued on opposite sides of the sample. The grids covered an area of 0.6 1 mm2 and were always placed in the center of the {110}-side faces of the Nb crystals [see Fig. 1(d)]. The {110}-side faces were chosen because the deformation of these side faces during the compression test was much less than that of the other side face of {100}- or {112}-type. This is caused by the crystallography of the operating slip systems, see Section 7.
Fig. 1. Sequence of the preparation of compression test specimens. (a) Stacking sequence for the bonding procedure, (b) UHV bonded sapphire/Nb/sapphire joint, (c) compression specimen with strain gauges (Ltot=5±8 mm; a,b = 3.820.3 mm; h = 1±3 mm), (d) position of the strain gauge on the side face (s1=0.6 mm; s2=1.0 mm).
SOYEZ et al.: CONSTRAINED YIELDING
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Table 2. Crystallographic orientation of the specimens and their fracture energies Gc [22] Specimen TCC1 TCC2 TCC3 TCC4
Orientation relationship
Fracture energy Gc [J/m2]
Nb(110)[001]6a-Al2O3(0001)[1120] Nb(110)[001]6a-Al2O3(0001)[1100] Nb(110)[001]6a-Al2O3(1100)[0001] Nb(111)[112]6a-Al2O3(0001)[1100]
1876 2 610 1899 2 213 114 2 25 112 2 51
3. MECHANICAL TESTING
The small specimen size required the construction of a special device for compression testing (Fig. 2). It consists of a hardened steel anvil on top of the sample and a steel end standard on the bottom anvil. The applied force was measured by a load cell with an accuracy of 0.1 N. A constant crosshead speed of 0.1 mm/min was used for the whole loading cycle. The load, the strain in the two strain gauges and the crosshead displacement were measured simultaneously as a function of time. The loading axis is perpendicular to the Nb/sapphire interfaces. All contact areas between device, anvil and specimen were lubricated with MoS2 to maintain a proper alignment of anvil and specimen. By using two strain gauges on opposite sides of the sample, the in¯uence of the bending of the sample was minimized. This was tested by loading the sample in the elastic region and comparing the measured strain of the two strain gauges, DMS1 and DMS2. If the dierence between DMS1 and DMS2 was larger than 25%, the alignment was checked and readjusted. A better alignment could not be achieved because of the small size of the samples. Figure 3 shows a measured load±strain-curve and the resulting average strain eavg=12 (eDMS1+eDMS2) of the sample. The compression test was terminated when a crack in the sapphire appeared indicated by a load drop. From the load±strain-curve the modulus of
Fig. 3. Example of a load±strain-curve. The strains of the two strain gauges DMS1 and DMS2 and the average strain eavg=12(eDMS1+eDMS2) are shown.
elasticity, the yield stress of the metal, the fracture stress of the joint and the strain at which fracture occurred were determined. 4. EVALUATION ALGORITHM FOR THE STRESS± STRAIN DATA
The ®rst experimental results showed that the values for the modulus of elasticity and the yield stress are strongly dependent on the evaluation of the load±strain-data. To calculate the stress in the loading direction, the measured load was divided by the initial cross section of the specimen. The change of the cross section during the compression test was neglected. The average strain eavg=12(eDMS1+eDMS2) was determined and afterwards the stress plotted against eavg. To ensure an equal evaluation for all load±strain-data, the measurements were treated in the same standardized method: First the elastic modulus was determined from the linear part of the stress±strain curve. In a second step, the yield stress sys0.02 at 0.02% strain was determined using the previously obtained elastic modulus. The applied procedure will be explained in more detail in the following paragraph. If the measured curve consists of n0 data points (xi, yi), a linear regression was performed for the interval [1; n0] and the linear regression coecient n0 X yi ÿ y x i ÿ x i1 s R s n0 n0 X X 2 x i ÿ x yi ÿ y2 i1
Fig. 2. Layout of the device for compression testing. The sapphire/Nb/sapphire joint is loaded perpendicular to the metal/ceramic interface.
1
i1
was calculated. The upper border of the interval was gradually reduced until the linear regression coecient was larger than a critical value Rc (see Fig. 4), which was arbitrarily chosen to be 99.9%. The slope of the line is then equal to the modulus of elasticity. Since for small metal thicknesses the strain prior to failure was less than 0.2% in some
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 4. Evaluation of the stress±strain data for a sandwich sample. The inset diagram is a magni®cation of the region where the transition from elasticity to plasticity takes place.
cases, the yield stress was measured at 0.02% strain instead of the typically used 0.2%. The yield stress sys0.02 is given where the stress±strain curve intersects a straight line having the slope of the elastic modulus and an oset of +0.02% (see Fig. 4). 5. RESULTS OF THE MECHANICAL TESTING
The stress±strain curves for three dierent metal thicknesses are shown in Fig. 5. The arrows mark the load drop by crack formation in sapphire. After cracking, the test was interrupted. With decreasing metal thickness the slope in the elastic region of the curves and also the plastic hardening of the samples increases. For metal layer thicknesses larger than 2 mm, the strain of the sample prior to failure sometimes exceeded the maximum applicable strain of the strain gauges of 3%. If this was the case, the loading cycle was interrupted and a second pair of strain gauges was glued on the sample. Afterwards the compression test was continued. An example
Fig. 5. Stress±strain curves for sandwich compression specimens TCC1 as a function of metal layer thickness h. The arrows indicate the stress drop at the failure of sapphire.
Fig. 6. Modulus of elasticity of specimens TCC1 to TCC4 as a function of metal layer thickness h. The error bars indicate the standard deviation obtained for compression specimens manufactured from one welded sample.
where a stress±strain-curve is measured in two cycles is shown in curve III of Fig. 5. The elastic behavior of the joints is plotted in Fig. 6. Since the grid of the strain gauge was smaller than the Nb side face, the strain was only measured for the metal. The yield stress at 0.02% strain was evaluated from the experimental stress±strain-curves as described in Section 4. The results for four dierent OR are plotted in Fig. 7, in which all plots are scaled equally for better comparison. For TCC1 and TCC3 the yield stress depends strongly on the metal thickness and varies between 45 MPa for thick metal layers and 100 MPa for thin ones. The combinations TCC2 and TCC4 show a less distinct dependency. The yield stress varies between 45 MPa and 65 MPa for TCC2 and between 50 MPa and 85 MPa for TCC4. 6. SEM STUDIES
The aim of the SEM studies was to obtain information about the slip mechanisms in the metal layer of the joints. For some samples side faces of {100}Nb-type in TCC1 to TCC3 and {112}Nb-type in TCC4 along with the adjacent {110}Nb-face in all cases were polished and investigated after the compression tests. On the {110}Nb- side faces slip traces are visible near the interface which discharge into the bright band B parallel to the interface (see Fig. 8). Only for strains larger than 2% can the band B be observed over the whole width of the Nb layer without interruption and also independent of the OR. The spacing between the band B and the interface varies between 1.5 mm and 5 mm with a mean value of about 2.5 mm. The slip traces inclined to the band B cannot be associated with low indexed directions in the Nb. They occur on the {110}Nb-side faces only in a small region of from 10 to 50 mm adjacent to the metal/ceramic interface.
SOYEZ et al.: CONSTRAINED YIELDING
3575
Fig. 7. Yield stress of specimens TCC1 to TCC4 as a function of the metal layer thickness.
No slip traces were observed in the region between the band B and the interface. In comparison, a great many slip traces could be seen on the {100}Nb-side face (Fig. 9). The very
wavy slip traces could be detected up to a distance of 300 mm from the interface and are typical for the pencil-glide in b.c.c.-metals [17]. With increasing distance from the interface, their density decreases.
Fig. 8. SEM-picture of the (110)Nb-side face at the Nb/sapphire (S)-interface region after 2.8% strain. Visible is a band (B) in the Nb at a distance of about 3 mm from the interface and the slip traces merging with that.
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 9. SEM-picture of the (001)Nb-side face after 1.8% strain in the vicinity of the Nb/sapphire (S) interface. The slip lines are very wavy and their density decreases with increasing distance from the interface.
SOYEZ et al.: CONSTRAINED YIELDING
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Fig. 10. Slip systems in Nb single crystals of {112}- and {110}-type. The arrows on the (110)-side-faces indicate the directions of the Burgers vectors.
The observations on the adjacent side faces of the Nb thus indicate that the main deformation takes place in the region up to 300 mm from the interface. A plain identi®cation of the activated slip systems using the slip traces on the polished surfaces was not possible.
ical model will be used to explain the observed relationship. This two-dimensional model is sucient since lateral elongation occurs only in one direction owing to the slip systems operating in Nb: the orientation of Nb in the joints leads to primary slip systems with Burgers vectors parallel to the (110)side face (Fig. 10). An outward ¯ow of these (110)-
7. INTERPRETATION AND DISCUSSION
The increase of the eective elastic modulus with decreasing metal thickness is due to the constraint acting on Nb at the metal/ceramic interface caused by the dierent Poisson's ratios. The in¯uence of the constraint increases with decreasing metal layer thickness. The values for samples with a metal layer thickness of 3 mm are the same as for bulk Nb, whereas the eective elastic modulus reaches values up to 300 GPa for the smaller metal thicknesses. By using Hooke's law for a cubic crystal in a triaxial stress state and assuming that there is no elongation of the sapphire perpendicular to the loading direction, we evaluate an eective modulus Ee for thin metal thicknesses (see Appendix A). The calculated Ee of 241 GPa agrees well with the measured values varying between 210 GPa and 295 GPa. This estimate illustrates, that a multiaxial stress state is produced as a result of the adhesion between Nb and sapphire. The experimental results give evidence of a relation between metal layer thickness and measured yield stress. A two-dimensional continuum mechan-
Fig. 11. Continuum mechanical model of the ¯ow behavior of sapphire/Nb/sapphire compression specimens. The volume element on the right of width dx and height h shows the acting stresses. C: ceramic; M: metal.
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 12. Yield stress sys0.02 as a function of a/h for specimens of orientation TCC1 to TCC3. Values of k and s0 for the plotted lines. (a) s0=29.9 MPa, k = 30.1 MPa. (b) s0=25.8 MPa, k = 27.1 MPa. (c) s0=21.7 MPa, k = 24.1 MPa.
side faces is thus not expected. As shown by stylus measurements, only a slight barrelling due to the multiaxial stress state near the metal/ceramic-interface was observed. The stresses acting on a volume element of the metal layer within the ceramic/metal/ceramic composite are shown in Fig. 11. Lateral expansion of Nb is impeded by the strong adhesion at the metal/ ceramic-interface which can be described by a shear stress k in x-direction. The traction k acting on the volume element is balanced by gradients of average normal stress sx. The absolute value of k is assumed to be constant along the interface. Force equilibrium in the x-direction leads to: X Fi 0, 2 i
)sx h ÿ sx dsx h ÿ 2 k dx 0:
3
This equation together with Trescas yield criterion [18, 19] and the boundary condition that the yield stress at the edge has to be the uniaxial yield stress result in the following expression for the local yield stress sz of the volume element:
Fig. 13. Yield stress sys0.02 as a function of a/h for specimens of orientation TCC4. Values of k and s0 for the plotted lines. (a) s0=58.1 MPa, k = 34.2 MPa. (b) k = 21.8 MPa. (c) s0=23.9 MPa, s0=41.0 MPa, k = 9.4 MPa.
sz s0 k
a ÿ 2x : h
4
Integration over the whole bond area leads to an equation for the eective yield stress as a function of the metal layer thickness for a sample width b: a=2 F b2 sz dx: 5 0
)
F k a sys0:02 s0 : A0 2 h
6
Equation (6) shows that the yield stress is inversely proportional to the metal layer thickness h. By plotting the eective yield stress against a/h (Figs 12 and 13) we obtain a linear relationship and the parameters s0 and k can be calculated. The values for s0 and k are listed in Table 3. Additionally, the uniaxial yield stress of the Nb single crystals was determined. Specimens of the same material and in the same orientation as in the sandwiches TCC1 to TCC3 were tested in compression with the loading axis parallel to the [110]Nb-direction. Lubricated platens were used. The samples were squares in cross section and had a/h-ratios from 0.63 to 2.47. The yield stress was independent of the a/h-ratio and had a mean value
Table 3. Results of the linear regression for the orientation combinations TCC1 to TCC4. Listed are the simulation parameters s0 and k [see equation (6)] and the regression coecient R [see equation (1)] Orientation relationship
s0 (MPa)
k (MPa)
R (%)
TCC1: Nb(110)[001]6a-Al2O3(0001)[1120] TCC2: Nb(110)[001]6a-Al2O3(0001)[1100] TCC3: Nb(110)[001]6a-Al2O3(1100)[0001] Mean of TCC1 to TCC3 and Standard Deviation TCC4: Nb(111)[112]6a-Al2O3(0001)[1100]
24.0 25.8 33.7 26.1 19.1 24.3 25.8 24.1 41.02 17.1
30.0 23.8 17.2 24.8 33.0 23.8 27.1 23.0 21.82 12.4
96 85 99 78
SOYEZ et al.: CONSTRAINED YIELDING
of 23.4 2 3.1 MPa. By using Schmid's law with a factor of 0.471 appropriate to this case, we ®nd a critical resolved shear stress tcrss of 11.0 2 1.5 MPa. The value for s0 of the metal/ceramic-joints (see Table 3) is consistent with the value obtained for uniaxial yield stress of the bulk Nb samples. Duesberry and Foxall [20] investigated high purity Nb single crystals in tension and compression and measured a critical resolved shear stress of 10 MPa. With a resulting Schmid factor of ms,[110]=0.471 and ms,[111]=0.314, the uniaxial yield stress for the [110]- and [111]-orientated Nb crystals is 21 MPa and 32 MPa, respectively. According to the model described above, the corresponding values are 25.8 2 4.1 MPa and 41.0 2 17.1 MPa which are in reasonable agreement with the calculated yield stress s0 of the bulk material. The shear stress k imposing the constraint on the lateral deformation of the Nb varies between 17 2 5 MPa and 33 24 MPa (see Table 3). Applying Trescas yield condition to a single crystal is a very rough approximation. A more detailed analysis considering the stresses acting on the dierent slip systems of {112}h111i- and {110}h111i-type shows that the shear stress k is equal to 3tcrss [21]. The obtained values of k lead to tcrss between 6 MPa and 11 MPa which agree well with the critical resolved shear stress tcrss 1 10 MPa [20] of high purity Nb. However, even by simply using Trescas yield condition, the mechanical behavior of the joints can be described reasonably well. Additionally, the stress state at the interface region is aected by the dierent Poisson's ratios of Nb and a-Al2O3 which result in compressive stresses parallel to the interface. Furthermore, due to the mismatch of the thermal expansion coecients there exist stresses at the interface that develop during diusion bonding. A quantitative determination of these dierent stresses acting at the interface was not possible and they are therefore neglected. Yielding parallel to the interface can readily be observed on SEM images (Section 6). Slip traces on the (110)-side face discharge into the band B. The lack of observation of other slip traces shows that mainly slip systems of the {112}Nbh111i- and {110}Nbh111i-type are activated. Glide processes parallel to the interface were also identi®ed by Burgers vector analysis in the TEM [22]. Slip traces B at a distance of from 2 mm to 3 mm from the interface become visible only after a certain level of deformation. This could be shown by SEM investigation of the side faces in dierent compression states [22]. In the area between the band B and the metal/ceramic interface, no other slip traces could be found. This indicates, that slip of dislocations from the bulk into region near to the interface is prevented. The change of direction of the slip traces could be explained by cross-slip mechanisms, which are easily possible in b.c.c.
3579
metals. By etching an undeformed sample, it was con®rmed that this region at the interface is already present before deformation occurs [22]. The formation of the band B can be explained by an increase of the yield stress at the near interface region due to an uptake of oxygen during diusion bonding. The high bonding temperature of 14008C leads to an dissolution of Al2O3 into the Nb near the interface [23]. The content decreases with the distance from the interface. It is well established that small amounts of O result in an increase of the shear stress near the interface [24, 25]. Korn [26] estimated the oxygen content in the interface region of similarly prepared Nb/sapphire joints by measuring the Vickers hardness on Nb fracture surfaces of four-point bend specimens using a method described by Elssner [24] and Szkopiak [25]. At bonding conditions of 14008C/3 h Korn found an oxygen content of 0.14 at.% in contrast to the content of unbonded single crystals of 0.028 at.%. The increase in yield stress and embrittlement with increasing oxygen content of Nb is also con®rmed by measurements of van Thorne [27]. He studied the in¯uence of the impurity content on the yield stress in Nb and found a linear increase of the yield stress with increasing impurity content. An in¯uence of interface strength on the mechanical behavior of Nb/sapphire joints in compression could not be observed. Of prime importance are the high fracture energies Gc, which even for the weaker joints are on the order of about 100 J/m2. The adhesion at the interfaces studied is so high, that yielding in the Nb parallel to the interface is energetically favorable over dislocation movement in the interface or detachment at the sapphire/Nb bond. The high interface strength between Nb and sapphire is also re¯ected in the failure of the joint through crack formation in the sapphire.
8. SUMMARY
Reproducible measurements of the mechanical properties of metal/ceramic joints can be performed using the described compression device. Elastic and plastic behavior of the joints is strongly in¯uenced by the constraint of the ceramic acting on the metal due to adhesion at the interface. This is demonstrated by the in¯uence of metal thicknesses on the eective Young's modulus and yield stress of the joints. For the Nb/sapphire system, an in¯uence of varying bond strength on the yield stress could not be observed. A continuum mechanical model was developed to describe the relation between the measured yield stress of the joint and the metal layer thickness. This model uses the uniaxial yield stress and the shear stress acting on the Nb parallel to the interface as parameters. It explains the exper-
3580
SOYEZ et al.: CONSTRAINED YIELDING
imentally observed relationship between yield stress and metal thickness. The yielding process parallel to the interface is con®rmed by the slip traces observed on the polished side faces of the compression samples.
30. Wachtman, J. B. Jr., Tet, W. E., Lam, D. G. Jr. and Stinch®eld, R. P., J. Res. Natl. Bur. Std., 1960, 64A, 213.
APPENDIX A AcknowledgementsÐThe authors would like to thank D. Conway at Cornell University (Cornell) and R. M. McMeeking at University of California (Santa Barbara) for helpful discussions.
According to Hooke's law, the strain for a cubic crystal in a three-dimensional principal stress state can be expressed by: e1
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1 21 31 s1 ÿ s2 ÿ s3 , E2 E3 E1
A:1a
e2 ÿ
12 1 32 s1 s2 ÿ s3 , E1 E3 E2
A:1b
e3 ÿ
13 23 1 s1 ÿ s2 s3 : E1 E2 E3
A:1c
The strains ei are functions of the Poisson's ratios nij, the moduli of elasticity Ei and the normal stresses si. Due to the cubic symmetry, only three of the elastic constants are linearly independent. To estimate the in¯uence of the adhesion at the metal/ceramic interface on the elastic behavior for thin metal layers, a-Al2O3 is assumed to be rigid. With the loading axis parallel to the e3-direction, the strains e1 and e2 are thus zero: e1 e2 0:
A:2
By using this assumption, a relation between s3 and e3 results that is only dependent on the Poisson's ratios nij and the modulus of elasticity E3. s3
E3 1 ÿ 12 21 e3 Eeff e3 f
A:3
where f = 1 ÿ n12n21 ÿ n13n31 ÿ n23n32 ÿ n13n21n32 ÿ n31n12n23. The values for the nij can be calculated from the stiness matrix S of niobium [28] using a method described by Turley and Sines [29]. The general system of coordinates is now assigned to the orientation of the Nb single crystal in the specimens TCC1 to TCC3 (Fig. A1). Then the Poisson's ratios nij for Nb can be calculated to be: 12 32 0:232; 13 31 0:59; 21 23 0:361: These values for the dierent nij and the modulus of elasticity E3=E[110]=93 GPa lead to an eective modulus Ee=241 GPa. To estimate the resulting error according to equation (A.2), the elongation of the sapphire perpendicular to the loading axis and thus the elongation of the Nb is taken as sapphire e e1 e2 ÿ s3 : A:4 Esapphire Inserting equation (A.4) in equations (A.1a)±(c) yields a modi®ed relation between s3 and e3 in comparison to equation (A.3):
Fig. A1. The general system of coordinates is assigned to the orientation of the Nb single crystal (TCC1 to TCC3).
SOYEZ et al.: CONSTRAINED YIELDING s3 E3 1 ÿ 12 21 e3 f 13 23 12 23 31 21 sapphire =Esapphire E3 E* e3 :
A:5
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By using the nij for Nb given above and the Poisson's ratio and the elastic modulus for sapphire nsapphire(0001)=0.17 and Esapphire[0001]=460 GPa [30] a value of 200 GPa is obtained for E*. The resulting error of Ee is in the order of 20%.
Acta mater. Vol. 46, No. 10, pp. 3571±3581, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00028-7 1359-6454/98 $19.00 + 0.00
CONSTRAINED YIELDING IN NIOBIUM SINGLE CRYSTALS BONDED TO SAPPHIRE G. SOYEZ1, G. ELSSNER1, M. RUÈHLE1 and R. RAJ2 Max-Planck-Institut fuÈr Metallforschung, D-70174 Stuttgart, Germany and 2University of Colorado, Boulder, CO 80309-0427, U.S.A.
1
(Received 20 October 1997; accepted 30 December 1997) AbstractÐThe mechanical properties of single crystalline sapphire/Nb/sapphire joints manufactured by diusion bonding in UHV are investigated for dierent orientation relationships (OR) between metal and ceramic at the interface. Compression tests resulted in the Young's moduli and the yield stresses of the joints for four dierent OR and metal thicknesses between 1 mm and 3 mm, respectively. The mechanical properties depend strongly on the thickness of the metal sheet, whereas the OR and hence the adhesion at the interface seem to have no signi®cant eect on the mechanical properties of the Nb/a-Al2O3 interface. The observed relationship between yield stress and metal thickness can be explained by a simple continuum mechanical model. # 1998 Acta Metallurgica Inc.
1. INTRODUCTION
The properties of metal/ceramic composites and joints are strongly in¯uenced by their heterophase interfaces. Interface failure depends on the elastic properties of the bonded materials, the thickness of the metal layer, the geometry of the specimens and the loading mode. Additionally, stresses caused by elastic and/or thermal mis®t lower the stability against mechanical loading. For optimized interfaces these residual stresses should be minimized. Information on the fracture energy and other measures of the bond strength prevail in the hitherto published studies on mechanical properties of metal/ceramic composites. These properties were investigated in general as a function of the material combination [1] and in the case of bicrystals as a function of the crystallographic orientation and the impurity content of the interface [2]. The dislocation mechanisms of plastic deformation near metal/ceramic interfaces however were not yet investigated. The experimental investigation of the mechanical behavior of metal/ceramic interfaces requires a wellde®ned preparation of the joints. In order to investigate the in¯uence of dierent OR at the interface on the mechanical properties, diusion bonding in an ultra high vacuum (UHV) was chosen for manufacturing. The model system niobium/sapphire was used since both materials possess a similar thermal expansion coecient and there are no intermediate reaction layers. Furthermore, the interface between Nb and sapphire is well characterized regarding structure [3±7] and mechanical properties [8±12]. In the present work the in¯uence of metal layer thickness, crystallographic orientation of the interface and fracture energy on the yield stress of the
metal part of the sapphire/Nb/sapphire sandwich was studied in compression tests. A second paper to be published will deal with the fracture mechanism of these sandwiches tested in compression. 2. SPECIMEN PREPARATION
High purity Nb and sapphire single crystals (impurity contents are listed in Table 1) were diusion bonded in UHV using a special apparatus described elsewhere [13]. To achieve optimum bonding conditions, the crystals have to be prepared in the following manner. After crystallographic orientation by Laue techniques, the surface planes of the crystals to be joined are prepared by grinding and mechanical polishing. The mean roughness of the polished surfaces is 10±20 nm for Nb and 8±15 nm for sapphire. Prior to inserting into the UHV bonding machine, the Nb is etched in concentrated HF. Afterwards both the metal and ceramic pieces are cleaned in acetone (p.a.) and methanol (p.a.) followed by ultrasonic agitation to remove remaining impurities from the polishing and etching process. The specimens are then dried in pure nitrogen (99.99 wt%). Before welding the surfaces to be bonded are sputter-cleaned by high-energetic Ar ions. The surface contamination was monitored by Auger electron spectroscopy. The stacking sequence of the sandwich con®guration and the subsequent preparation steps are shown in Fig. 1. Preliminary experiments showed, that stress concentrations between sapphire and alumina at the edges and at the contact areas, respectively, resulted in cracking of the sapphire during the bonding procedure. To avoid cracking,
3571
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SOYEZ et al.: CONSTRAINED YIELDING Table 1. Impurity contents of the single crystals used
Nb Sapphire (a-Al2O3)
50 wt±ppm O2 R100 ppm Mg
27 wt±ppm N2 R20 ppm Ca
the sapphire/Nb/sapphire specimen was enclosed between two Mo platelets. Thus stress concentrations acting on the sapphire are reduced by the plasticity of the Mo. Furthermore, Mo does not bond with the Al2O3. The whole stack [Fig. 1(a)] is heated to a bonding temperature of 14008C within 1.5 h, held at this temperature for 3 h and then cooled down to room temperature in another 1.5 h. After reaching the bonding temperature of 14008C a pressure of about 7 MPa was applied to the stack of specimens. The single crystals in the stack were orientated so that the same OR holds at both Nb/sapphire interfaces, see Table 2. Dierent interface strengths, characterized by the fracture energy Gc in fourpoint bending, were realized by changing the orientation of the sapphire with respect to the Nb. In the ®rst three tri-crystal-combinations TCC1 to TCC3, the {110}Nb-plane is bonded onto sapphire planes with dierent OR. The identical orientation of the Nb should guarantee that the activated slip systems and hence the deformation mode should be the same for TCC1 to TCC3. TCC2 is consistent with the OR found in internally oxidized Nb±Alalloys [14]. TCC4 is dierent from the ®rst three
combinations since a (111)Nb-plane is bonded onto (0001)-sapphire substrates. TCC4 corresponds to the OR obtained for MBE grown overlayers on (0001) sapphire [15, 16]. The bonded stack [Fig. 1(b)] is withdrawn from the UHV diusion bonding machine and cut into several compression test specimens of the type shown in Fig. 1(c). The limited size of the samples that can be handled within the UHV diusion bonding machine and the cross section of the Nb single crystals determined the dimensions of the compression specimens. Up to four compression specimens could be cut out from one diusion bonded sample. They had a length between 5 mm and 8 mm and a quadratic cross section with a specimen width a 1 b of 3.8 2 0.3 mm. To modify the in¯uence of the interface strength on the mechanical properties, the thickness h of the Nb layer was varied between 1 mm and 3 mm. A parallel alignment of the top and bottom surfaces was attained by grinding. To measure the strain, two strain gauges were glued on opposite sides of the sample. The grids covered an area of 0.6 1 mm2 and were always placed in the center of the {110}-side faces of the Nb crystals [see Fig. 1(d)]. The {110}-side faces were chosen because the deformation of these side faces during the compression test was much less than that of the other side face of {100}- or {112}-type. This is caused by the crystallography of the operating slip systems, see Section 7.
Fig. 1. Sequence of the preparation of compression test specimens. (a) Stacking sequence for the bonding procedure, (b) UHV bonded sapphire/Nb/sapphire joint, (c) compression specimen with strain gauges (Ltot=5±8 mm; a,b = 3.820.3 mm; h = 1±3 mm), (d) position of the strain gauge on the side face (s1=0.6 mm; s2=1.0 mm).
SOYEZ et al.: CONSTRAINED YIELDING
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Table 2. Crystallographic orientation of the specimens and their fracture energies Gc [22] Specimen TCC1 TCC2 TCC3 TCC4
Orientation relationship
Fracture energy Gc [J/m2]
Nb(110)[001]6a-Al2O3(0001)[1120] Nb(110)[001]6a-Al2O3(0001)[1100] Nb(110)[001]6a-Al2O3(1100)[0001] Nb(111)[112]6a-Al2O3(0001)[1100]
1876 2 610 1899 2 213 114 2 25 112 2 51
3. MECHANICAL TESTING
The small specimen size required the construction of a special device for compression testing (Fig. 2). It consists of a hardened steel anvil on top of the sample and a steel end standard on the bottom anvil. The applied force was measured by a load cell with an accuracy of 0.1 N. A constant crosshead speed of 0.1 mm/min was used for the whole loading cycle. The load, the strain in the two strain gauges and the crosshead displacement were measured simultaneously as a function of time. The loading axis is perpendicular to the Nb/sapphire interfaces. All contact areas between device, anvil and specimen were lubricated with MoS2 to maintain a proper alignment of anvil and specimen. By using two strain gauges on opposite sides of the sample, the in¯uence of the bending of the sample was minimized. This was tested by loading the sample in the elastic region and comparing the measured strain of the two strain gauges, DMS1 and DMS2. If the dierence between DMS1 and DMS2 was larger than 25%, the alignment was checked and readjusted. A better alignment could not be achieved because of the small size of the samples. Figure 3 shows a measured load±strain-curve and the resulting average strain eavg=12 (eDMS1+eDMS2) of the sample. The compression test was terminated when a crack in the sapphire appeared indicated by a load drop. From the load±strain-curve the modulus of
Fig. 3. Example of a load±strain-curve. The strains of the two strain gauges DMS1 and DMS2 and the average strain eavg=12(eDMS1+eDMS2) are shown.
elasticity, the yield stress of the metal, the fracture stress of the joint and the strain at which fracture occurred were determined. 4. EVALUATION ALGORITHM FOR THE STRESS± STRAIN DATA
The ®rst experimental results showed that the values for the modulus of elasticity and the yield stress are strongly dependent on the evaluation of the load±strain-data. To calculate the stress in the loading direction, the measured load was divided by the initial cross section of the specimen. The change of the cross section during the compression test was neglected. The average strain eavg=12(eDMS1+eDMS2) was determined and afterwards the stress plotted against eavg. To ensure an equal evaluation for all load±strain-data, the measurements were treated in the same standardized method: First the elastic modulus was determined from the linear part of the stress±strain curve. In a second step, the yield stress sys0.02 at 0.02% strain was determined using the previously obtained elastic modulus. The applied procedure will be explained in more detail in the following paragraph. If the measured curve consists of n0 data points (xi, yi), a linear regression was performed for the interval [1; n0] and the linear regression coecient n0 X yi ÿ y x i ÿ x i1 s R s n0 n0 X X 2 x i ÿ x yi ÿ y2 i1
Fig. 2. Layout of the device for compression testing. The sapphire/Nb/sapphire joint is loaded perpendicular to the metal/ceramic interface.
1
i1
was calculated. The upper border of the interval was gradually reduced until the linear regression coecient was larger than a critical value Rc (see Fig. 4), which was arbitrarily chosen to be 99.9%. The slope of the line is then equal to the modulus of elasticity. Since for small metal thicknesses the strain prior to failure was less than 0.2% in some
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 4. Evaluation of the stress±strain data for a sandwich sample. The inset diagram is a magni®cation of the region where the transition from elasticity to plasticity takes place.
cases, the yield stress was measured at 0.02% strain instead of the typically used 0.2%. The yield stress sys0.02 is given where the stress±strain curve intersects a straight line having the slope of the elastic modulus and an oset of +0.02% (see Fig. 4). 5. RESULTS OF THE MECHANICAL TESTING
The stress±strain curves for three dierent metal thicknesses are shown in Fig. 5. The arrows mark the load drop by crack formation in sapphire. After cracking, the test was interrupted. With decreasing metal thickness the slope in the elastic region of the curves and also the plastic hardening of the samples increases. For metal layer thicknesses larger than 2 mm, the strain of the sample prior to failure sometimes exceeded the maximum applicable strain of the strain gauges of 3%. If this was the case, the loading cycle was interrupted and a second pair of strain gauges was glued on the sample. Afterwards the compression test was continued. An example
Fig. 5. Stress±strain curves for sandwich compression specimens TCC1 as a function of metal layer thickness h. The arrows indicate the stress drop at the failure of sapphire.
Fig. 6. Modulus of elasticity of specimens TCC1 to TCC4 as a function of metal layer thickness h. The error bars indicate the standard deviation obtained for compression specimens manufactured from one welded sample.
where a stress±strain-curve is measured in two cycles is shown in curve III of Fig. 5. The elastic behavior of the joints is plotted in Fig. 6. Since the grid of the strain gauge was smaller than the Nb side face, the strain was only measured for the metal. The yield stress at 0.02% strain was evaluated from the experimental stress±strain-curves as described in Section 4. The results for four dierent OR are plotted in Fig. 7, in which all plots are scaled equally for better comparison. For TCC1 and TCC3 the yield stress depends strongly on the metal thickness and varies between 45 MPa for thick metal layers and 100 MPa for thin ones. The combinations TCC2 and TCC4 show a less distinct dependency. The yield stress varies between 45 MPa and 65 MPa for TCC2 and between 50 MPa and 85 MPa for TCC4. 6. SEM STUDIES
The aim of the SEM studies was to obtain information about the slip mechanisms in the metal layer of the joints. For some samples side faces of {100}Nb-type in TCC1 to TCC3 and {112}Nb-type in TCC4 along with the adjacent {110}Nb-face in all cases were polished and investigated after the compression tests. On the {110}Nb- side faces slip traces are visible near the interface which discharge into the bright band B parallel to the interface (see Fig. 8). Only for strains larger than 2% can the band B be observed over the whole width of the Nb layer without interruption and also independent of the OR. The spacing between the band B and the interface varies between 1.5 mm and 5 mm with a mean value of about 2.5 mm. The slip traces inclined to the band B cannot be associated with low indexed directions in the Nb. They occur on the {110}Nb-side faces only in a small region of from 10 to 50 mm adjacent to the metal/ceramic interface.
SOYEZ et al.: CONSTRAINED YIELDING
3575
Fig. 7. Yield stress of specimens TCC1 to TCC4 as a function of the metal layer thickness.
No slip traces were observed in the region between the band B and the interface. In comparison, a great many slip traces could be seen on the {100}Nb-side face (Fig. 9). The very
wavy slip traces could be detected up to a distance of 300 mm from the interface and are typical for the pencil-glide in b.c.c.-metals [17]. With increasing distance from the interface, their density decreases.
Fig. 8. SEM-picture of the (110)Nb-side face at the Nb/sapphire (S)-interface region after 2.8% strain. Visible is a band (B) in the Nb at a distance of about 3 mm from the interface and the slip traces merging with that.
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 9. SEM-picture of the (001)Nb-side face after 1.8% strain in the vicinity of the Nb/sapphire (S) interface. The slip lines are very wavy and their density decreases with increasing distance from the interface.
SOYEZ et al.: CONSTRAINED YIELDING
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Fig. 10. Slip systems in Nb single crystals of {112}- and {110}-type. The arrows on the (110)-side-faces indicate the directions of the Burgers vectors.
The observations on the adjacent side faces of the Nb thus indicate that the main deformation takes place in the region up to 300 mm from the interface. A plain identi®cation of the activated slip systems using the slip traces on the polished surfaces was not possible.
ical model will be used to explain the observed relationship. This two-dimensional model is sucient since lateral elongation occurs only in one direction owing to the slip systems operating in Nb: the orientation of Nb in the joints leads to primary slip systems with Burgers vectors parallel to the (110)side face (Fig. 10). An outward ¯ow of these (110)-
7. INTERPRETATION AND DISCUSSION
The increase of the eective elastic modulus with decreasing metal thickness is due to the constraint acting on Nb at the metal/ceramic interface caused by the dierent Poisson's ratios. The in¯uence of the constraint increases with decreasing metal layer thickness. The values for samples with a metal layer thickness of 3 mm are the same as for bulk Nb, whereas the eective elastic modulus reaches values up to 300 GPa for the smaller metal thicknesses. By using Hooke's law for a cubic crystal in a triaxial stress state and assuming that there is no elongation of the sapphire perpendicular to the loading direction, we evaluate an eective modulus Ee for thin metal thicknesses (see Appendix A). The calculated Ee of 241 GPa agrees well with the measured values varying between 210 GPa and 295 GPa. This estimate illustrates, that a multiaxial stress state is produced as a result of the adhesion between Nb and sapphire. The experimental results give evidence of a relation between metal layer thickness and measured yield stress. A two-dimensional continuum mechan-
Fig. 11. Continuum mechanical model of the ¯ow behavior of sapphire/Nb/sapphire compression specimens. The volume element on the right of width dx and height h shows the acting stresses. C: ceramic; M: metal.
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SOYEZ et al.: CONSTRAINED YIELDING
Fig. 12. Yield stress sys0.02 as a function of a/h for specimens of orientation TCC1 to TCC3. Values of k and s0 for the plotted lines. (a) s0=29.9 MPa, k = 30.1 MPa. (b) s0=25.8 MPa, k = 27.1 MPa. (c) s0=21.7 MPa, k = 24.1 MPa.
side faces is thus not expected. As shown by stylus measurements, only a slight barrelling due to the multiaxial stress state near the metal/ceramic-interface was observed. The stresses acting on a volume element of the metal layer within the ceramic/metal/ceramic composite are shown in Fig. 11. Lateral expansion of Nb is impeded by the strong adhesion at the metal/ ceramic-interface which can be described by a shear stress k in x-direction. The traction k acting on the volume element is balanced by gradients of average normal stress sx. The absolute value of k is assumed to be constant along the interface. Force equilibrium in the x-direction leads to: X Fi 0, 2 i
)sx h ÿ sx dsx h ÿ 2 k dx 0:
3
This equation together with Trescas yield criterion [18, 19] and the boundary condition that the yield stress at the edge has to be the uniaxial yield stress result in the following expression for the local yield stress sz of the volume element:
Fig. 13. Yield stress sys0.02 as a function of a/h for specimens of orientation TCC4. Values of k and s0 for the plotted lines. (a) s0=58.1 MPa, k = 34.2 MPa. (b) k = 21.8 MPa. (c) s0=23.9 MPa, s0=41.0 MPa, k = 9.4 MPa.
sz s0 k
a ÿ 2x : h
4
Integration over the whole bond area leads to an equation for the eective yield stress as a function of the metal layer thickness for a sample width b: a=2 F b2 sz dx: 5 0
)
F k a sys0:02 s0 : A0 2 h
6
Equation (6) shows that the yield stress is inversely proportional to the metal layer thickness h. By plotting the eective yield stress against a/h (Figs 12 and 13) we obtain a linear relationship and the parameters s0 and k can be calculated. The values for s0 and k are listed in Table 3. Additionally, the uniaxial yield stress of the Nb single crystals was determined. Specimens of the same material and in the same orientation as in the sandwiches TCC1 to TCC3 were tested in compression with the loading axis parallel to the [110]Nb-direction. Lubricated platens were used. The samples were squares in cross section and had a/h-ratios from 0.63 to 2.47. The yield stress was independent of the a/h-ratio and had a mean value
Table 3. Results of the linear regression for the orientation combinations TCC1 to TCC4. Listed are the simulation parameters s0 and k [see equation (6)] and the regression coecient R [see equation (1)] Orientation relationship
s0 (MPa)
k (MPa)
R (%)
TCC1: Nb(110)[001]6a-Al2O3(0001)[1120] TCC2: Nb(110)[001]6a-Al2O3(0001)[1100] TCC3: Nb(110)[001]6a-Al2O3(1100)[0001] Mean of TCC1 to TCC3 and Standard Deviation TCC4: Nb(111)[112]6a-Al2O3(0001)[1100]
24.0 25.8 33.7 26.1 19.1 24.3 25.8 24.1 41.02 17.1
30.0 23.8 17.2 24.8 33.0 23.8 27.1 23.0 21.82 12.4
96 85 99 78
SOYEZ et al.: CONSTRAINED YIELDING
of 23.4 2 3.1 MPa. By using Schmid's law with a factor of 0.471 appropriate to this case, we ®nd a critical resolved shear stress tcrss of 11.0 2 1.5 MPa. The value for s0 of the metal/ceramic-joints (see Table 3) is consistent with the value obtained for uniaxial yield stress of the bulk Nb samples. Duesberry and Foxall [20] investigated high purity Nb single crystals in tension and compression and measured a critical resolved shear stress of 10 MPa. With a resulting Schmid factor of ms,[110]=0.471 and ms,[111]=0.314, the uniaxial yield stress for the [110]- and [111]-orientated Nb crystals is 21 MPa and 32 MPa, respectively. According to the model described above, the corresponding values are 25.8 2 4.1 MPa and 41.0 2 17.1 MPa which are in reasonable agreement with the calculated yield stress s0 of the bulk material. The shear stress k imposing the constraint on the lateral deformation of the Nb varies between 17 2 5 MPa and 33 24 MPa (see Table 3). Applying Trescas yield condition to a single crystal is a very rough approximation. A more detailed analysis considering the stresses acting on the dierent slip systems of {112}h111i- and {110}h111i-type shows that the shear stress k is equal to 3tcrss [21]. The obtained values of k lead to tcrss between 6 MPa and 11 MPa which agree well with the critical resolved shear stress tcrss 1 10 MPa [20] of high purity Nb. However, even by simply using Trescas yield condition, the mechanical behavior of the joints can be described reasonably well. Additionally, the stress state at the interface region is aected by the dierent Poisson's ratios of Nb and a-Al2O3 which result in compressive stresses parallel to the interface. Furthermore, due to the mismatch of the thermal expansion coecients there exist stresses at the interface that develop during diusion bonding. A quantitative determination of these dierent stresses acting at the interface was not possible and they are therefore neglected. Yielding parallel to the interface can readily be observed on SEM images (Section 6). Slip traces on the (110)-side face discharge into the band B. The lack of observation of other slip traces shows that mainly slip systems of the {112}Nbh111i- and {110}Nbh111i-type are activated. Glide processes parallel to the interface were also identi®ed by Burgers vector analysis in the TEM [22]. Slip traces B at a distance of from 2 mm to 3 mm from the interface become visible only after a certain level of deformation. This could be shown by SEM investigation of the side faces in dierent compression states [22]. In the area between the band B and the metal/ceramic interface, no other slip traces could be found. This indicates, that slip of dislocations from the bulk into region near to the interface is prevented. The change of direction of the slip traces could be explained by cross-slip mechanisms, which are easily possible in b.c.c.
3579
metals. By etching an undeformed sample, it was con®rmed that this region at the interface is already present before deformation occurs [22]. The formation of the band B can be explained by an increase of the yield stress at the near interface region due to an uptake of oxygen during diusion bonding. The high bonding temperature of 14008C leads to an dissolution of Al2O3 into the Nb near the interface [23]. The content decreases with the distance from the interface. It is well established that small amounts of O result in an increase of the shear stress near the interface [24, 25]. Korn [26] estimated the oxygen content in the interface region of similarly prepared Nb/sapphire joints by measuring the Vickers hardness on Nb fracture surfaces of four-point bend specimens using a method described by Elssner [24] and Szkopiak [25]. At bonding conditions of 14008C/3 h Korn found an oxygen content of 0.14 at.% in contrast to the content of unbonded single crystals of 0.028 at.%. The increase in yield stress and embrittlement with increasing oxygen content of Nb is also con®rmed by measurements of van Thorne [27]. He studied the in¯uence of the impurity content on the yield stress in Nb and found a linear increase of the yield stress with increasing impurity content. An in¯uence of interface strength on the mechanical behavior of Nb/sapphire joints in compression could not be observed. Of prime importance are the high fracture energies Gc, which even for the weaker joints are on the order of about 100 J/m2. The adhesion at the interfaces studied is so high, that yielding in the Nb parallel to the interface is energetically favorable over dislocation movement in the interface or detachment at the sapphire/Nb bond. The high interface strength between Nb and sapphire is also re¯ected in the failure of the joint through crack formation in the sapphire.
8. SUMMARY
Reproducible measurements of the mechanical properties of metal/ceramic joints can be performed using the described compression device. Elastic and plastic behavior of the joints is strongly in¯uenced by the constraint of the ceramic acting on the metal due to adhesion at the interface. This is demonstrated by the in¯uence of metal thicknesses on the eective Young's modulus and yield stress of the joints. For the Nb/sapphire system, an in¯uence of varying bond strength on the yield stress could not be observed. A continuum mechanical model was developed to describe the relation between the measured yield stress of the joint and the metal layer thickness. This model uses the uniaxial yield stress and the shear stress acting on the Nb parallel to the interface as parameters. It explains the exper-
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SOYEZ et al.: CONSTRAINED YIELDING
imentally observed relationship between yield stress and metal thickness. The yielding process parallel to the interface is con®rmed by the slip traces observed on the polished side faces of the compression samples.
30. Wachtman, J. B. Jr., Tet, W. E., Lam, D. G. Jr. and Stinch®eld, R. P., J. Res. Natl. Bur. Std., 1960, 64A, 213.
APPENDIX A AcknowledgementsÐThe authors would like to thank D. Conway at Cornell University (Cornell) and R. M. McMeeking at University of California (Santa Barbara) for helpful discussions.
According to Hooke's law, the strain for a cubic crystal in a three-dimensional principal stress state can be expressed by: e1
REFERENCES 1. Elssner, G. and Petzow, G., ISIJ Int., 1990, 30, 1011. 2. Korn, D., Elssner, G., Fischmeister, H. F. and RuÈhle, M., Acta metall. mater., 1992, 40, S355. 3. Florjancic, M., Mader, W., RuÈhle, M. and Turwitt, M., J. Phys., 1985, C 4, 129. 4. RuÈhle, M., Backhaus-Ricoult, M., Burger, K. and Mader, W., in Ceramic Microstructures, ed. Pask and Evans. Plenum Publishing Coorperation, 1988, p. 195. 5. Mader, W. and RuÈhle, M., Acta metall. mater., 1989, 37, 853. 6. Bruley, J., Brydson, R., MuÈllejans, H., Mayer, J., Gutekunst, G., Mader, W., Knauss, D. and RuÈhle, M., J. Mater. Res., 1994, 9, 2574. 7. Mayer, J., Gutekunst, G., MoÈbus, G., Dura, J., Flynn, C. P. and RuÈhle, M., Acta metall. mater., 1992, 40, S217. 8. Evans, A. G., RuÈhle, M. and Turwitt, M., J. Phys., 1985, C 4, 613. 9. Gibbesch, B. and Elssner, G., Acta metall. mater., 1992, 40, S59. 10. O'Dowd, N. P., Stout, M. G. and Shih, C. F., Philos. Mag., 1992, A 66, 1037. 11. Wu, X., Holmes, J. W. and Ghosh, A. K., Acta metall. mater., 1994, 42, 2069. 12. Evans, A. G., Lu, M. C., Schmauder, S. and RuÈhle, M., Acta metall. mater., 1986, 34, 1643. 13. Fischmeister, H. F., Elssner, G., Gibbesch, B., Kadow, K.-H., Kawa, F., Korn, D., Mader, W. and Turwitt, M., Rev. Sci. Instrum., 1993, 64, 234. 14. Mader, W., Mater. Res. Soc. Symp. Proc., 1987, 82, 403. 15. Durbin, S. M., Cunningham, J. E. and Flynn, C. P., J. Phys. F: Met. Phys., 1982, 12, L75. 16. Mayer, J., Flynn, C. P. and RuÈhle, M., Ultramicroscopy, 1990, 33, 51. 17. Bowen, D. K., Christian, J. W. and Taylor, G., Can. J. Phys., 1967, 45, 903. 18. Tresca, H., C.R. Acad. Sci., 1864, 59, 754. 19. Courtney, T. H., Mechanical Behaviour of Materials. McGraw±Hill, 1990, 17. 20. Duesbery, M. S. and Foxall, R. A., Philos. Mag, 1969, 20, 719. 21. unpublished results. 22. Soyez, G., Ph.D. thesis, Stuttgart, 1996. 23. Elssner, G. and Petzow, G., Z. Metallkd., 1973, 64, 280. 24. Elssner, G. and HoÈrz, G., Z. Metallkd., 1971, 62, 217. 25. Szkopiak, Z. C., J. Less-Common Metals, 1969, 19, 93. 26. Korn, D., VDI Fortschrittsberichte Reihe 5, Nr. 320. VDI±Verlag, 1993. 27. van Thorne, L. I. and Thomas, G., Acta metall., 1963, 11, 881. 28. Carroll, K. J., J. Appl. Phys., 1965, 36, 3689. 29. Turley, J. and Sines, G., J. Phys. D: Appl. Phys., 1971, 4, 264.
1 21 31 s1 ÿ s2 ÿ s3 , E2 E3 E1
A:1a
e2 ÿ
12 1 32 s1 s2 ÿ s3 , E1 E3 E2
A:1b
e3 ÿ
13 23 1 s1 ÿ s2 s3 : E1 E2 E3
A:1c
The strains ei are functions of the Poisson's ratios nij, the moduli of elasticity Ei and the normal stresses si. Due to the cubic symmetry, only three of the elastic constants are linearly independent. To estimate the in¯uence of the adhesion at the metal/ceramic interface on the elastic behavior for thin metal layers, a-Al2O3 is assumed to be rigid. With the loading axis parallel to the e3-direction, the strains e1 and e2 are thus zero: e1 e2 0:
A:2
By using this assumption, a relation between s3 and e3 results that is only dependent on the Poisson's ratios nij and the modulus of elasticity E3. s3
E3 1 ÿ 12 21 e3 Eeff e3 f
A:3
where f = 1 ÿ n12n21 ÿ n13n31 ÿ n23n32 ÿ n13n21n32 ÿ n31n12n23. The values for the nij can be calculated from the stiness matrix S of niobium [28] using a method described by Turley and Sines [29]. The general system of coordinates is now assigned to the orientation of the Nb single crystal in the specimens TCC1 to TCC3 (Fig. A1). Then the Poisson's ratios nij for Nb can be calculated to be: 12 32 0:232; 13 31 0:59; 21 23 0:361: These values for the dierent nij and the modulus of elasticity E3=E[110]=93 GPa lead to an eective modulus Ee=241 GPa. To estimate the resulting error according to equation (A.2), the elongation of the sapphire perpendicular to the loading axis and thus the elongation of the Nb is taken as sapphire e e1 e2 ÿ s3 : A:4 Esapphire Inserting equation (A.4) in equations (A.1a)±(c) yields a modi®ed relation between s3 and e3 in comparison to equation (A.3):
Fig. A1. The general system of coordinates is assigned to the orientation of the Nb single crystal (TCC1 to TCC3).
SOYEZ et al.: CONSTRAINED YIELDING s3 E3 1 ÿ 12 21 e3 f 13 23 12 23 31 21 sapphire =Esapphire E3 E* e3 :
A:5
3581
By using the nij for Nb given above and the Poisson's ratio and the elastic modulus for sapphire nsapphire(0001)=0.17 and Esapphire[0001]=460 GPa [30] a value of 200 GPa is obtained for E*. The resulting error of Ee is in the order of 20%.